Geometric Median-Shift over Riemannian Manifolds

Abstract

Manifold clustering finds wide applications in many areas. In this paper, we propose a new kernel function that makes use of Riemannian geodesic distances among data points, and present a Geometric median shift algorithm over Riemannian Manifolds. Relying on the geometric median shift, together with geodesic distances, our approach is able to effectively cluster data points distributed over Riemannian manifolds. In addition to improving the clustering results, the complexity for calculating geometric median is reduced to O(n ²), compared to O(n ²logn ²) for Tukey median. Using both Riemannian Manifolds and Euclidean spaces, we compare the geometric median shift and mean shift algorithms for clustering synthetic and real data sets.